Input values to these routines may be any scalar number or string that looks
like a number and represents an integer.

•

Leading and trailing whitespace is ignored.

•

Leading and trailing zeros are ignored.

•

If the string has a "0x" prefix, it is interpreted as a
hexadecimal number.

•

If the string has a "0b" prefix, it is interpreted as a binary
number.

•

One underline is allowed between any two digits.

•

If the string can not be interpreted, NaN is returned.

Octal numbers are typically prefixed by "0", but since leading zeros
are stripped, these methods can not automatically recognize octal numbers, so
use the constructor from_oct() to interpret octal strings.

Each of the methods below (except config(), accuracy() and
precision()) accepts three additional parameters. These arguments $A,
$P and $R are "accuracy", "precision" and
"round_mode". Please see the section about "ACCURACY and
PRECISION" for more information.

Setting a class variable effects all object instance that are created
afterwards.

Set or get the precision, i.e., the place to round relative to the decimal
point. The precision must be a integer. Setting the precision to $P means
that each number is rounded up or down, depending on the rounding mode, to
the nearest multiple of 10**$P. If the precision is set to
"undef", no rounding is done.

You might want to use " accuracy()" instead. With
"accuracy()" you set the number of digits each result
should have, with " precision()" you set the place where
to round.

Please see the section about "ACCURACY and PRECISION" for further
details.

Note: Each class has its own globals separated from Math::BigInt, but it is
possible to subclass Math::BigInt and make the globals of the subclass
aliases to the ones from Math::BigInt.

div_scale()

Set/get the fallback accuracy. This is the accuracy used when neither
accuracy nor precision is set explicitly. It is used when a computation
might otherwise attempt to return an infinite number of digits.

round_mode()

Set/get the rounding mode.

upgrade()

Set/get the class for upgrading. When a computation might result in a
non-integer, the operands are upgraded to this class. This is used for
instance by bignum. The default is "undef", thus the following
operation creates a Math::BigInt, not a Math::BigFloat:

Creates a new Math::BigInt object from a scalar or another Math::BigInt
object. The input is accepted as decimal, hexadecimal (with leading '0x')
or binary (with leading '0b').

See "Input" for more info on accepted input formats.

from_hex()

$x = Math::BigInt->from_hex("0xcafe"); # input is hexadecimal

Interpret input as a hexadecimal string. A "0x" or "x"
prefix is optional. A single underscore character may be placed right
after the prefix, if present, or between any two digits. If the input is
invalid, a NaN is returned.

from_oct()

$x = Math::BigInt->from_oct("0775"); # input is octal

Interpret the input as an octal string and return the corresponding value. A
"0" (zero) prefix is optional. A single underscore character may
be placed right after the prefix, if present, or between any two digits.
If the input is invalid, a NaN is returned.

from_bin()

$x = Math::BigInt->from_bin("0b10011"); # input is binary

Interpret the input as a binary string. A "0b" or "b"
prefix is optional. A single underscore character may be placed right
after the prefix, if present, or between any two digits. If the input is
invalid, a NaN is returned.

from_bytes()

$x = Math::BigInt->from_bytes("\xf3\x6b"); # $x = 62315

Interpret the input as a byte string, assuming big endian byte order. The
output is always a non-negative, finite integer.

In some special cases, from_bytes() matches the conversion done by
unpack():

Given a string, a base, and an optional collation sequence, interpret the
string as a number in the given base. The collation sequence describes the
value of each character in the string.

If a collation sequence is not given, a default collation sequence is used.
If the base is less than or equal to 36, the collation sequence is the
string consisting of the 36 characters "0" to "9" and
"A" to "Z". In this case, the letter case in the input
is ignored. If the base is greater than 36, and smaller than or equal to
62, the collation sequence is the string consisting of the 62 characters
"0" to "9", "A" to "Z", and
"a" to "z". A base larger than 62 requires the
collation sequence to be specified explicitly.

These examples show standard binary, octal, and hexadecimal conversion. All
cases return 250.

Creates a new Math::BigInt object representing one. The optional argument is
either '-' or '+', indicating whether you want plus one or minus one. If
used as an instance method, assigns the value to the invocand.

binf()

$x = Math::BigInt->binf($sign);

Creates a new Math::BigInt object representing infinity. The optional
argument is either '-' or '+', indicating whether you want infinity or
minus infinity. If used as an instance method, assigns the value to the
invocand.

$x->binf();
$x->binf('-');

bnan()

$x = Math::BigInt->bnan();

Creates a new Math::BigInt object representing NaN (Not A Number). If used
as an instance method, assigns the value to the invocand.

$x->bnan();

bpi()

$x = Math::BigInt->bpi(100); # 3
$x->bpi(100); # 3

Creates a new Math::BigInt object representing PI. If used as an instance
method, assigns the value to the invocand. With Math::BigInt this always
returns 3.

If upgrading is in effect, returns PI, rounded to N digits with the current
rounding mode:

These methods are called when Math::BigInt encounters an object it doesn't
know how to handle. For instance, assume $x is a Math::BigInt, or subclass
thereof, and $y is defined, but not a Math::BigInt, or subclass thereof.
If you do

$x -> badd($y);

$y needs to be converted into an object that $x can deal with. This is done
by first checking if $y is something that $x might be upgraded to. If that
is the case, no further attempts are made. The next is to see if $y
supports the method "as_int()". If it does, "as_int()"
is called, but if it doesn't, the next thing is to see if $y supports the
method "as_number()". If it does, "as_number()" is
called. The method "as_int()" (and "as_number()") is
expected to return either an object that has the same class as $x, a
subclass thereof, or a string that "ref($x)->new()" can parse
to create an object.

"as_number()" is an alias to "as_int()".
"as_number" was introduced in v1.22, while "as_int()"
was introduced in v1.68.

Negate the number, e.g. change the sign between '+' and '-', or between
'+inf' and '-inf', respectively. Does nothing for NaN or zero.

babs()

$x->babs();

Set the number to its absolute value, e.g. change the sign from '-' to '+'
and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
numbers.

bsgn()

$x->bsgn();

Signum function. Set the number to -1, 0, or 1, depending on whether the
number is negative, zero, or positive, respectively. Does not modify
NaNs.

bnorm()

$x->bnorm(); # normalize (no-op)

Normalize the number. This is a no-op and is provided only for backwards
compatibility.

binc()

$x->binc(); # increment x by 1

bdec()

$x->bdec(); # decrement x by 1

badd()

$x->badd($y); # addition (add $y to $x)

bsub()

$x->bsub($y); # subtraction (subtract $y from $x)

bmul()

$x->bmul($y); # multiplication (multiply $x by $y)

bmuladd()

$x->bmuladd($y,$z);

Multiply $x by $y, and then add $z to the result,

This method was added in v1.87 of Math::BigInt (June 2007).

bdiv()

$x->bdiv($y); # divide, set $x to quotient

Divides $x by $y by doing floored division (F-division), where the quotient
is the floored (rounded towards negative infinity) quotient of the two
operands. In list context, returns the quotient and the remainder. The
remainder is either zero or has the same sign as the second operand. In
scalar context, only the quotient is returned.

The quotient is always the greatest integer less than or equal to the
real-valued quotient of the two operands, and the remainder (when it is
non-zero) always has the same sign as the second operand; so, for example,

The behavior of the overloaded operator % agrees with the behavior of Perl's
built-in % operator (as documented in the perlop manpage), and the
equation

$x == ($x / $y) * $y + ($x % $y)

holds true for any finite $x and finite, non-zero $y.

Perl's "use integer" might change the behaviour of % and / for
scalars. This is because under 'use integer' Perl does what the underlying
C library thinks is right, and this varies. However, "use
integer" does not change the way things are done with Math::BigInt
objects.

btdiv()

$x->btdiv($y); # divide, set $x to quotient

Divides $x by $y by doing truncated division (T-division), where quotient is
the truncated (rouneded towards zero) quotient of the two operands. In
list context, returns the quotient and the remainder. The remainder is
either zero or has the same sign as the first operand. In scalar context,
only the quotient is returned.

Returns the value of $num taken to the power $exp in the modulus $mod using
binary exponentiation. "bmodpow" is far superior to writing

$num ** $exp % $mod

because it is much faster - it reduces internal variables into the modulus
whenever possible, so it operates on smaller numbers.

"bmodpow" also supports negative exponents.

bmodpow($num, -1, $mod)

is exactly equivalent to

bmodinv($num, $mod)

bpow()

$x->bpow($y); # power of arguments (x ** y)

"bpow()" (and the rounding functions) now modifies the first
argument and returns it, unlike the old code which left it alone and only
returned the result. This is to be consistent with "badd()" etc.
The first three modifies $x, the last one won't:

Calculates the binomial coefficient n over k, also called the
"choose" function, which is

( n ) n!
| | = --------
( k ) k!(n-k)!

when n and k are non-negative. This method implements the full Kronenburg
extension (Kronenburg, M.J. "The Binomial Coefficient for Negative
Arguments." 18 May 2011. http://arxiv.org/abs/1105.3689/) illustrated
by the following pseudo-code:

Returns the factorial of $x, i.e., the product of all positive integers up
to and including $x.

bdfac()

$x->bdfac(); # double factorial of $x (1*2*3*4*..*$x)

Returns the double factorial of $x. If $x is an even integer, returns the
product of all positive, even integers up to and including $x, i.e.,
2*4*6*...*$x. If $x is an odd integer, returns the product of all
positive, odd integers, i.e., 1*3*5*...*$x.

Returns the number of digits in the decimal representation of the number. In
list context, returns the length of the integer and fraction part. For
Math::BigInt objects, the length of the fraction part is always 0.

The following probably doesn't do what you expect:

$c = Math::BigInt->new(123);
print $c->length(),"\n"; # prints 30

It prints both the number of digits in the number and in the fraction part
since print calls "length()" in list context. Use something
like:

print scalar $c->length(),"\n"; # prints 3

mantissa()

$x->mantissa();

Return the signed mantissa of $x as a Math::BigInt.

exponent()

$x->exponent();

Return the exponent of $x as a Math::BigInt.

parts()

$x->parts();

Returns the significand (mantissa) and the exponent as integers. In
Math::BigFloat, both are returned as Math::BigInt objects.

sparts()

Returns the significand (mantissa) and the exponent as integers. In scalar
context, only the significand is returned. The significand is the integer
with the smallest absolute value. The output of "sparts()"
corresponds to the output from "bsstr()".

In Math::BigInt, this method is identical to "parts()".

nparts()

Returns the significand (mantissa) and exponent corresponding to
normalized notation. In scalar context, only the significand is returned.
For finite non-zero numbers, the significand's absolute value is greater
than or equal to 1 and less than 10. The output of "nparts()"
corresponds to the output from "bnstr()". In Math::BigInt, if
the significand can not be represented as an integer, upgrading is
performed or NaN is returned.

eparts()

Returns the significand (mantissa) and exponent corresponding to
engineering notation. In scalar context, only the significand is returned.
For finite non-zero numbers, the significand's absolute value is greater
than or equal to 1 and less than 1000, and the exponent is a multiple of
3. The output of "eparts()" corresponds to the output from
"bestr()". In Math::BigInt, if the significand can not be
represented as an integer, upgrading is performed or NaN is returned.

dparts()

Returns the integer part and the fraction part. If the fraction part can
not be represented as an integer, upgrading is performed or NaN is
returned. The output of "dparts()" corresponds to the output
from "bdstr()".

Returns a string representing the number using decimal notation. In
Math::BigFloat, the output is zero padded according to the current
accuracy or precision, if any of those are defined.

bsstr()

Returns a string representing the number using scientific notation where
both the significand (mantissa) and the exponent are integers. The output
corresponds to the output from "sparts()".

123 is returned as "123e+0"
1230 is returned as "123e+1"
12300 is returned as "123e+2"
12000 is returned as "12e+3"
10000 is returned as "1e+4"

bnstr()

Returns a string representing the number using normalized notation, the
most common variant of scientific notation. For finite non-zero numbers,
the absolute value of the significand is greater than or equal to 1 and
less than 10. The output corresponds to the output from
"nparts()".

123 is returned as "1.23e+2"
1230 is returned as "1.23e+3"
12300 is returned as "1.23e+4"
12000 is returned as "1.2e+4"
10000 is returned as "1e+4"

bestr()

Returns a string representing the number using engineering notation. For
finite non-zero numbers, the absolute value of the significand is greater
than or equal to 1 and less than 1000, and the exponent is a multiple of
3. The output corresponds to the output from "eparts()".

123 is returned as "123e+0"
1230 is returned as "1.23e+3"
12300 is returned as "12.3e+3"
12000 is returned as "12e+3"
10000 is returned as "10e+3"

bdstr()

Returns a string representing the number using decimal notation. The
output corresponds to the output from "dparts()".

123 is returned as "123"
1230 is returned as "1230"
12300 is returned as "12300"
12000 is returned as "12000"
10000 is returned as "10000"

to_hex()

$x->to_hex();

Returns a hexadecimal string representation of the number. See also
from_hex().

to_bin()

$x->to_bin();

Returns a binary string representation of the number. See also
from_bin().

to_oct()

$x->to_oct();

Returns an octal string representation of the number. See also
from_oct().

Precision is a fixed number of digits before (positive) or after (negative) the
decimal point. For example, 123.45 has a precision of -2. 0 means an integer
like 123 (or 120). A precision of 2 means at least two digits to the left of
the decimal point are zero, so 123 with P = 1 becomes 120. Note that numbers
with zeros before the decimal point may have different precisions, because
1200 can have P = 0, 1 or 2 (depending on what the initial value was). It
could also have p < 0, when the digits after the decimal point are zero.

Number of significant digits. Leading zeros are not counted. A number may have
an accuracy greater than the non-zero digits when there are zeros in it or
trailing zeros. For example, 123.456 has A of 6, 10203 has 5, 123.0506 has 7,
123.45000 has 8 and 0.000123 has 3.

When rounding a number, different 'styles' or 'kinds' of rounding are possible.
(Note that random rounding, as in Math::Round, is not implemented.)

Directed rounding

These round modes always round in the same direction.

'trunc'

Round towards zero. Remove all digits following the rounding place, i.e.,
replace them with zeros. Thus, 987.65 rounded to tens (P=1) becomes 980,
and rounded to the fourth significant digit becomes 987.6 (A=4). 123.456
rounded to the second place after the decimal point (P=-2) becomes 123.46.
This corresponds to the IEEE 754 rounding mode 'roundTowardZero'.

Rounding to nearest

These rounding modes round to the nearest digit. They differ in how they
determine which way to round in the ambiguous case when there is a tie.

Round away from zero, i.e., round to the number with the largest absolute
value. E.g., when rounding to the nearest integer, -1.5 becomes -2, 1.5
becomes 2 and 1.49 becomes 1. This corresponds to the IEEE 754 rounding
mode 'roundTiesToAway'.

The handling of A & P in MBI/MBF (the old core code shipped with Perl
versions <= 5.7.2) is like this:

Precision

* bfround($p) is able to round to $p number of digits after the decimal
point
* otherwise P is unused

Accuracy (significant digits)

* bround($a) rounds to $a significant digits
* only bdiv() and bsqrt() take A as (optional) parameter
+ other operations simply create the same number (bneg etc), or
more (bmul) of digits
+ rounding/truncating is only done when explicitly calling one
of bround or bfround, and never for Math::BigInt (not implemented)
* bsqrt() simply hands its accuracy argument over to bdiv.
* the documentation and the comment in the code indicate two
different ways on how bdiv() determines the maximum number
of digits it should calculate, and the actual code does yet
another thing
POD:
max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
Comment:
result has at most max(scale, length(dividend), length(divisor)) digits
Actual code:
scale = max(scale, length(dividend)-1,length(divisor)-1);
scale += length(divisor) - length(dividend);
So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10
So for lx = 3, ly = 9, scale = 10, scale will actually be 16
(10+9-3). Actually, the 'difference' added to the scale is cal-
culated from the number of "significant digits" in dividend and
divisor, which is derived by looking at the length of the man-
tissa. Which is wrong, since it includes the + sign (oops) and
actually gets 2 for '+100' and 4 for '+101'. Oops again. Thus
124/3 with div_scale=1 will get you '41.3' based on the strange
assumption that 124 has 3 significant digits, while 120/7 will
get you '17', not '17.1' since 120 is thought to have 2 signif-
icant digits. The rounding after the division then uses the
remainder and $y to determine whether it must round up or down.
? I have no idea which is the right way. That's why I used a slightly more
? simple scheme and tweaked the few failing testcases to match it.

This is how it works now:

Setting/Accessing

* You can set the A global via Math::BigInt->accuracy() or
Math::BigFloat->accuracy() or whatever class you are using.
* You can also set P globally by using Math::SomeClass->precision()
likewise.
* Globals are classwide, and not inherited by subclasses.
* to undefine A, use Math::SomeCLass->accuracy(undef);
* to undefine P, use Math::SomeClass->precision(undef);
* Setting Math::SomeClass->accuracy() clears automatically
Math::SomeClass->precision(), and vice versa.
* To be valid, A must be > 0, P can have any value.
* If P is negative, this means round to the P'th place to the right of the
decimal point; positive values mean to the left of the decimal point.
P of 0 means round to integer.
* to find out the current global A, use Math::SomeClass->accuracy()
* to find out the current global P, use Math::SomeClass->precision()
* use $x->accuracy() respective $x->precision() for the local
setting of $x.
* Please note that $x->accuracy() respective $x->precision()
return eventually defined global A or P, when $x's A or P is not
set.

Creating numbers

* When you create a number, you can give the desired A or P via:
$x = Math::BigInt->new($number,$A,$P);
* Only one of A or P can be defined, otherwise the result is NaN
* If no A or P is give ($x = Math::BigInt->new($number) form), then the
globals (if set) will be used. Thus changing the global defaults later on
will not change the A or P of previously created numbers (i.e., A and P of
$x will be what was in effect when $x was created)
* If given undef for A and P, NO rounding will occur, and the globals will
NOT be used. This is used by subclasses to create numbers without
suffering rounding in the parent. Thus a subclass is able to have its own
globals enforced upon creation of a number by using
$x = Math::BigInt->new($number,undef,undef):
use Math::BigInt::SomeSubclass;
use Math::BigInt;
Math::BigInt->accuracy(2);
Math::BigInt::SomeSubClass->accuracy(3);
$x = Math::BigInt::SomeSubClass->new(1234);
$x is now 1230, and not 1200. A subclass might choose to implement
this otherwise, e.g. falling back to the parent's A and P.

Usage

* If A or P are enabled/defined, they are used to round the result of each
operation according to the rules below
* Negative P is ignored in Math::BigInt, since Math::BigInt objects never
have digits after the decimal point
* Math::BigFloat uses Math::BigInt internally, but setting A or P inside
Math::BigInt as globals does not tamper with the parts of a Math::BigFloat.
A flag is used to mark all Math::BigFloat numbers as 'never round'.

Precedence

* It only makes sense that a number has only one of A or P at a time.
If you set either A or P on one object, or globally, the other one will
be automatically cleared.
* If two objects are involved in an operation, and one of them has A in
effect, and the other P, this results in an error (NaN).
* A takes precedence over P (Hint: A comes before P).
If neither of them is defined, nothing is used, i.e. the result will have
as many digits as it can (with an exception for bdiv/bsqrt) and will not
be rounded.
* There is another setting for bdiv() (and thus for bsqrt()). If neither of
A or P is defined, bdiv() will use a fallback (F) of $div_scale digits.
If either the dividend's or the divisor's mantissa has more digits than
the value of F, the higher value will be used instead of F.
This is to limit the digits (A) of the result (just consider what would
happen with unlimited A and P in the case of 1/3 :-)
* bdiv will calculate (at least) 4 more digits than required (determined by
A, P or F), and, if F is not used, round the result
(this will still fail in the case of a result like 0.12345000000001 with A
or P of 5, but this can not be helped - or can it?)
* Thus you can have the math done by on Math::Big* class in two modi:
+ never round (this is the default):
This is done by setting A and P to undef. No math operation
will round the result, with bdiv() and bsqrt() as exceptions to guard
against overflows. You must explicitly call bround(), bfround() or
round() (the latter with parameters).
Note: Once you have rounded a number, the settings will 'stick' on it
and 'infect' all other numbers engaged in math operations with it, since
local settings have the highest precedence. So, to get SaferRound[tm],
use a copy() before rounding like this:
$x = Math::BigFloat->new(12.34);
$y = Math::BigFloat->new(98.76);
$z = $x * $y; # 1218.6984
print $x->copy()->bround(3); # 12.3 (but A is now 3!)
$z = $x * $y; # still 1218.6984, without
# copy would have been 1210!
+ round after each op:
After each single operation (except for testing like is_zero()), the
method round() is called and the result is rounded appropriately. By
setting proper values for A and P, you can have all-the-same-A or
all-the-same-P modes. For example, Math::Currency might set A to undef,
and P to -2, globally.
?Maybe an extra option that forbids local A & P settings would be in order,
?so that intermediate rounding does not 'poison' further math?

Overriding globals

* you will be able to give A, P and R as an argument to all the calculation
routines; the second parameter is A, the third one is P, and the fourth is
R (shift right by one for binary operations like badd). P is used only if
the first parameter (A) is undefined. These three parameters override the
globals in the order detailed as follows, i.e. the first defined value
wins:
(local: per object, global: global default, parameter: argument to sub)
+ parameter A
+ parameter P
+ local A (if defined on both of the operands: smaller one is taken)
+ local P (if defined on both of the operands: bigger one is taken)
+ global A
+ global P
+ global F
* bsqrt() will hand its arguments to bdiv(), as it used to, only now for two
arguments (A and P) instead of one

Local settings

* You can set A or P locally by using $x->accuracy() or
$x->precision()
and thus force different A and P for different objects/numbers.
* Setting A or P this way immediately rounds $x to the new value.
* $x->accuracy() clears $x->precision(), and vice versa.

Rounding

* the rounding routines will use the respective global or local settings.
bround() is for accuracy rounding, while bfround() is for precision
* the two rounding functions take as the second parameter one of the
following rounding modes (R):
'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
* you can set/get the global R by using Math::SomeClass->round_mode()
or by setting $Math::SomeClass::round_mode
* after each operation, $result->round() is called, and the result may
eventually be rounded (that is, if A or P were set either locally,
globally or as parameter to the operation)
* to manually round a number, call $x->round($A,$P,$round_mode);
this will round the number by using the appropriate rounding function
and then normalize it.
* rounding modifies the local settings of the number:
$x = Math::BigFloat->new(123.456);
$x->accuracy(5);
$x->bround(4);
Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
will be 4 from now on.

Default values

* R: 'even'
* F: 40
* A: undef
* P: undef

Remarks

* The defaults are set up so that the new code gives the same results as
the old code (except in a few cases on bdiv):
+ Both A and P are undefined and thus will not be used for rounding
after each operation.
+ round() is thus a no-op, unless given extra parameters A and P

Math with the numbers is done (by default) by a module called
"Math::BigInt::Calc". This is equivalent to saying:

use Math::BigInt try => 'Calc';

You can change this backend library by using:

use Math::BigInt try => 'GMP';

Note: General purpose packages should not be explicit about the library
to use; let the script author decide which is best.

If your script works with huge numbers and Calc is too slow for them, you can
also for the loading of one of these libraries and if none of them can be
used, the code dies:

use Math::BigInt only => 'GMP,Pari';

The following would first try to find Math::BigInt::Foo, then Math::BigInt::Bar,
and when this also fails, revert to Math::BigInt::Calc:

use Math::BigInt try => 'Foo,Math::BigInt::Bar';

The library that is loaded last is used. Note that this can be overwritten at
any time by loading a different library, and numbers constructed with
different libraries cannot be used in math operations together.

What library to use?

Note: General purpose packages should not be explicit about the library
to use; let the script author decide which is best.

Math::BigInt::GMP and Math::BigInt::Pari are in cases involving big numbers much
faster than Calc, however it is slower when dealing with very small numbers
(less than about 20 digits) and when converting very large numbers to decimal
(for instance for printing, rounding, calculating their length in decimal
etc).

So please select carefully what library you want to use.

Different low-level libraries use different formats to store the numbers.
However, you should NOT depend on the number having a specific format
internally.

See the respective math library module documentation for further details.

A sign of 'NaN' is used to represent the result when input arguments are not
numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
minus infinity. You get '+inf' when dividing a positive number by 0, and
'-inf' when dividing any negative number by 0.

Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
must be made in the second case. For long numbers, the copy can eat up to 20%
of the work (in the case of addition/subtraction, less for
multiplication/division). If $y is very small compared to $x, the form $x +=
$y is MUCH faster than $x = $x + $y since making the copy of $x takes more
time then the actual addition.

With a technique called copy-on-write, the cost of copying with overload could
be minimized or even completely avoided. A test implementation of COW did show
performance gains for overloaded math, but introduced a performance loss due
to a constant overhead for all other operations. So Math::BigInt does
currently not COW.

The rewritten version of this module (vs. v0.01) is slower on certain
operations, like "new()", "bstr()" and
"numify()". The reason are that it does now more work and handles
much more cases. The time spent in these operations is usually gained in the
other math operations so that code on the average should get (much) faster. If
they don't, please contact the author.

Some operations may be slower for small numbers, but are significantly faster
for big numbers. Other operations are now constant (O(1), like
"bneg()", "babs()" etc), instead of O(N) and thus nearly
always take much less time. These optimizations were done on purpose.

If you find the Calc module to slow, try to install any of the replacement
modules and see if they help you.

The basic design of Math::BigInt allows simple subclasses with very little work,
as long as a few simple rules are followed:

•

The public API must remain consistent, i.e. if a sub-class is overloading
addition, the sub-class must use the same name, in this case
badd(). The reason for this is that Math::BigInt is optimized to
call the object methods directly.

•

The private object hash keys like "$x->{sign}" may not be
changed, but additional keys can be added, like
"$x->{_custom}".

•

Accessor functions are available for all existing object hash keys and
should be used instead of directly accessing the internal hash keys. The
reason for this is that Math::BigInt itself has a pluggable interface
which permits it to support different storage methods.

More complex sub-classes may have to replicate more of the logic internal of
Math::BigInt if they need to change more basic behaviors. A subclass that
needs to merely change the output only needs to overload "bstr()".

All other object methods and overloaded functions can be directly inherited from
the parent class.

At the very minimum, any subclass needs to provide its own "new()" and
can store additional hash keys in the object. There are also some package
globals that must be defined, e.g.:

Additionally, you might want to provide the following two globals to allow
auto-upgrading and auto-downgrading to work correctly:

$upgrade = undef;
$downgrade = undef;

This allows Math::BigInt to correctly retrieve package globals from the
subclass, like $SubClass::precision. See t/Math/BigInt/Subclass.pm or
t/Math/BigFloat/SubClass.pm completely functional subclass examples.

Don't forget to

use overload;

in your subclass to automatically inherit the overloading from the parent. If
you like, you can change part of the overloading, look at Math::String for an
example.

Some things might not work as you expect them. Below is documented what is known
to be troublesome:

Comparing numbers as strings

Both "bstr()" and "bsstr()" as well as stringify via
overload drop the leading '+'. This is to be consistent with Perl and to
make "cmp" (especially with overloading) to work as you expect.
It also solves problems with "Test.pm" and Test::More, which
stringify arguments before comparing them.

Mark Biggar said, when asked about to drop the '+' altogether, or make only
"cmp" work:

I agree (with the first alternative), don't add the '+' on positive
numbers. It's not as important anymore with the new internal form
for numbers. It made doing things like abs and neg easier, but
those have to be done differently now anyway.

There is now a "bsstr()" method to get the string in scientific
notation aka 1e+2 instead of 100. Be advised that overloaded 'eq' always
uses bstr() for comparison, but Perl represents some numbers as 100
and others as 1e+308. If in doubt, convert both arguments to Math::BigInt
before comparing them as strings:

Alternatively, simply use "<=>" for comparisons, this always
gets it right. There is not yet a way to get a number automatically
represented as a string that matches exactly the way Perl represents it.

See also the section about "Infinity and Not a Number" for
problems in comparing NaNs.

int()

"int()" returns (at least for Perl v5.7.1 and up) another
Math::BigInt, not a Perl scalar:

This is seldom necessary, though, because this is done automatically, like
when you access an array:

$z = $array[$x]; # does work automatically

Modifying and =

Beware of:

$x = Math::BigFloat->new(5);
$y = $x;

This makes a second reference to the same object and stores it in $y.
Thus anything that modifies $x (except overloaded operators) also modifies
$y, and vice versa. Or in other words, "=" is only safe if you
modify your Math::BigInt objects only via overloaded math. As soon as you
use a method call it breaks:

$x->bmul(2);
print "$x, $y\n"; # prints '10, 10'

If you want a true copy of $x, use:

$y = $x->copy();

You can also chain the calls like this, this first makes a copy and then
multiply it by 2:

$y = $x->copy()->bmul(2);

See also the documentation for overload.pm regarding "=".

Overloading -$x

The following:

$x = -$x;

is slower than

$x->bneg();

since overload calls "sub($x,0,1);" instead of
"neg($x)". The first variant needs to preserve $x since it does
not know that it later gets overwritten. This makes a copy of $x and takes
O(N), but $x-> bneg() is O(1).

Mixing different object types

With overloaded operators, it is the first (dominating) operand that
determines which method is called. Here are some examples showing what
actually gets called in various cases.

For instance, Math::BigInt-> bdiv() always returns a Math::BigInt,
regardless of whether the second operant is a Math::BigFloat. To get a
Math::BigFloat you either need to call the operation manually, make sure
each operand already is a Math::BigFloat, or cast to that type via
Math::BigFloat-> new():

$float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5

Beware of casting the entire expression, as this would cast the result, at
which point it is too late:

Please report any bugs or feature requests to "bug-math-bigint at
rt.cpan.org", or through the web interface at
<https://rt.cpan.org/Ticket/Create.html?Queue=Math-BigInt> (requires
login). We will be notified, and then you'll automatically be notified of
progress on your bug as I make changes.